Calculus Examples

$f (x) = 5 x^{2}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 5 {(x + h)}^{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = 5 ((x + h) (x + h))$

Step 2.1.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

$f (x + h) = 5 (x (x + h) + h (x + h))$

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = 5 (x \cdot x + x h + h (x + h))$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = 5 (x \cdot x + x h + h x + h \cdot h)$

$f (x + h) = 5 (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 5 (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 5 (x^{2} + x h + h x + h^{2})$

$f (x + h) = 5 (x^{2} + x h + h x + h^{2})$

Step 2.1.2.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 5 (x^{2} + h x + h x + h^{2})$

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 5 (x^{2} + 2 h x + h^{2})$

$f (x + h) = 5 (x^{2} + 2 h x + h^{2})$

$f (x + h) = 5 (x^{2} + 2 h x + h^{2})$

Step 2.1.2.4

Apply the distributive property.

$f (x + h) = 5 x^{2} + 5 (2 h x) + 5 h^{2}$

Step 2.1.2.5

Multiply $2$ by $5$ .

$f (x + h) = 5 x^{2} + 10 h x + 5 h^{2}$

Step 2.1.2.6

The final answer is $5 x^{2} + 10 h x + 5 h^{2}$ .

$5 x^{2} + 10 h x + 5 h^{2}$

$5 x^{2} + 10 h x + 5 h^{2}$

$5 x^{2} + 10 h x + 5 h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $5 x^{2}$ .

$10 h x + 5 h^{2} + 5 x^{2}$

Step 2.2.2

Reorder $10 h x$ and $5 h^{2}$ .

$5 h^{2} + 10 h x + 5 x^{2}$

$5 h^{2} + 10 h x + 5 x^{2}$

Step 2.3

Find the components of the definition.

$f (x + h) = 5 h^{2} + 10 h x + 5 x^{2}$

$f (x) = 5 x^{2}$

$f (x + h) = 5 h^{2} + 10 h x + 5 x^{2}$

$f (x) = 5 x^{2}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{5 h^{2} + 10 h x + 5 x^{2} - (5 x^{2})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $5$ .

$\frac{5 h^{2} + 10 h x + 5 x^{2} - 5 x^{2}}{h}$

Step 4.1.2

Subtract $5 x^{2}$ from $5 x^{2}$ .

$\frac{5 h^{2} + 10 h x + 0}{h}$

Step 4.1.3

Add $5 h^{2} + 10 h x$ and $0$ .

$\frac{5 h^{2} + 10 h x}{h}$

Step 4.1.4

Factor $5 h$ out of $5 h^{2} + 10 h x$ .

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Step 4.1.4.1

Factor $5 h$ out of $5 h^{2}$ .

$\frac{5 h \cdot h + 10 h x}{h}$

Step 4.1.4.2

Factor $5 h$ out of $10 h x$ .

$\frac{5 h \cdot h + 5 h (2 x)}{h}$

Step 4.1.4.3

Factor $5 h$ out of $5 h \cdot h + 5 h (2 x)$ .

$\frac{5 h (h + 2 x)}{h}$

$\frac{5 h (h + 2 x)}{h}$

$\frac{5 h (h + 2 x)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 h (h + 2 x)}{h}$

Step 4.2.1.2

Divide $5 (h + 2 x)$ by $1$ .

$5 (h + 2 x)$

$5 (h + 2 x)$

Step 4.2.2

Apply the distributive property.

$5 h + 5 (2 x)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply $2$ by $5$ .

$5 h + 10 x$

Step 4.2.3.2

Reorder $5 h$ and $10 x$ .

$10 x + 5 h$

$10 x + 5 h$

$10 x + 5 h$

$10 x + 5 h$

Step 5

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$